# Properties

 Label 80.4.c.c Level $80$ Weight $4$ Character orbit 80.c Analytic conductor $4.720$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 80.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.72015280046$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b3 - b1 - 1) * q^5 + (-b2 + 2*b1) * q^7 + (-2*b3 - b2 - b1 - 1) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{3} - \beta_1 - 1) q^{5} + ( - \beta_{2} + 2 \beta_1) q^{7} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{9} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 20) q^{11} + ( - 7 \beta_{2} - 5 \beta_1) q^{13} + ( - 5 \beta_{2} - 10 \beta_1 + 20) q^{15} + (2 \beta_{2} + 18 \beta_1) q^{17} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 20) q^{19} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 68) q^{21} + (21 \beta_{2} + 8 \beta_1) q^{23} + (4 \beta_{3} + 15 \beta_{2} + 9 \beta_1 + 69) q^{25} + ( - 10 \beta_{2} + 8 \beta_1) q^{27} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 70) q^{29} + (24 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 96) q^{31} + ( - 20 \beta_{2} - 56 \beta_1) q^{33} + ( - 4 \beta_{3} - 20 \beta_{2} - 19 \beta_1 + 76) q^{35} + (21 \beta_{2} - 13 \beta_1) q^{37} + (24 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 56) q^{39} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 262) q^{41} + (18 \beta_{2} + 39 \beta_1) q^{43} + (3 \beta_{3} + 15 \beta_{2} + 8 \beta_1 + 193) q^{45} + (11 \beta_{2} - 54 \beta_1) q^{47} + (6 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 123) q^{49} + ( - 40 \beta_{3} - 20 \beta_{2} - 20 \beta_1 - 480) q^{51} + ( - 9 \beta_{2} + 101 \beta_1) q^{53} + (24 \beta_{3} + 30 \beta_{2} + 34 \beta_1 + 404) q^{55} + ( - 20 \beta_{2} - 16 \beta_1) q^{57} + ( - 44 \beta_{3} - 22 \beta_{2} - 22 \beta_1 + 348) q^{59} + (14 \beta_{3} + 7 \beta_{2} + 7 \beta_1 - 346) q^{61} + ( - 37 \beta_{2} - 32 \beta_1) q^{63} + ( - 28 \beta_{3} - 45 \beta_{2} + 57 \beta_1 + 152) q^{65} + ( - 18 \beta_{2} - 25 \beta_1) q^{67} + ( - 58 \beta_{3} - 29 \beta_{2} - 29 \beta_1 + 28) q^{69} + ( - 16 \beta_{3} - 8 \beta_{2} - 8 \beta_1 - 344) q^{71} + (64 \beta_{2} - 28 \beta_1) q^{73} + ( - 40 \beta_{3} + 85 \beta_1 - 40) q^{75} + ( - 56 \beta_{2} - 100 \beta_1) q^{77} + (8 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 368) q^{79} + ( - 50 \beta_{3} - 25 \beta_{2} - 25 \beta_1 - 371) q^{81} + ( - 114 \beta_{2} - 65 \beta_1) q^{83} + (8 \beta_{3} - 70 \beta_{2} - 182 \beta_1 + 288) q^{85} + (40 \beta_{2} + 142 \beta_1) q^{87} + (24 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 330) q^{89} + (80 \beta_{3} + 40 \beta_{2} + 40 \beta_1 - 248) q^{91} + (120 \beta_{2} + 120 \beta_1) q^{93} + ( - 16 \beta_{3} + 30 \beta_{2} - 6 \beta_1 + 364) q^{95} + (46 \beta_{2} + 86 \beta_1) q^{97} + (44 \beta_{3} + 22 \beta_{2} + 22 \beta_1 + 788) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b3 - b1 - 1) * q^5 + (-b2 + 2*b1) * q^7 + (-2*b3 - b2 - b1 - 1) * q^9 + (-4*b3 - 2*b2 - 2*b1 - 20) * q^11 + (-7*b2 - 5*b1) * q^13 + (-5*b2 - 10*b1 + 20) * q^15 + (2*b2 + 18*b1) * q^17 + (-4*b3 - 2*b2 - 2*b1 + 20) * q^19 + (-2*b3 - b2 - b1 - 68) * q^21 + (21*b2 + 8*b1) * q^23 + (4*b3 + 15*b2 + 9*b1 + 69) * q^25 + (-10*b2 + 8*b1) * q^27 + (8*b3 + 4*b2 + 4*b1 + 70) * q^29 + (24*b3 + 12*b2 + 12*b1 - 96) * q^31 + (-20*b2 - 56*b1) * q^33 + (-4*b3 - 20*b2 - 19*b1 + 76) * q^35 + (21*b2 - 13*b1) * q^37 + (24*b3 + 12*b2 + 12*b1 + 56) * q^39 + (6*b3 + 3*b2 + 3*b1 - 262) * q^41 + (18*b2 + 39*b1) * q^43 + (3*b3 + 15*b2 + 8*b1 + 193) * q^45 + (11*b2 - 54*b1) * q^47 + (6*b3 + 3*b2 + 3*b1 + 123) * q^49 + (-40*b3 - 20*b2 - 20*b1 - 480) * q^51 + (-9*b2 + 101*b1) * q^53 + (24*b3 + 30*b2 + 34*b1 + 404) * q^55 + (-20*b2 - 16*b1) * q^57 + (-44*b3 - 22*b2 - 22*b1 + 348) * q^59 + (14*b3 + 7*b2 + 7*b1 - 346) * q^61 + (-37*b2 - 32*b1) * q^63 + (-28*b3 - 45*b2 + 57*b1 + 152) * q^65 + (-18*b2 - 25*b1) * q^67 + (-58*b3 - 29*b2 - 29*b1 + 28) * q^69 + (-16*b3 - 8*b2 - 8*b1 - 344) * q^71 + (64*b2 - 28*b1) * q^73 + (-40*b3 + 85*b1 - 40) * q^75 + (-56*b2 - 100*b1) * q^77 + (8*b3 + 4*b2 + 4*b1 - 368) * q^79 + (-50*b3 - 25*b2 - 25*b1 - 371) * q^81 + (-114*b2 - 65*b1) * q^83 + (8*b3 - 70*b2 - 182*b1 + 288) * q^85 + (40*b2 + 142*b1) * q^87 + (24*b3 + 12*b2 + 12*b1 + 330) * q^89 + (80*b3 + 40*b2 + 40*b1 - 248) * q^91 + (120*b2 + 120*b1) * q^93 + (-16*b3 + 30*b2 - 6*b1 + 364) * q^95 + (46*b2 + 86*b1) * q^97 + (44*b3 + 22*b2 + 22*b1 + 788) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{5} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^5 - 4 * q^9 $$4 q - 4 q^{5} - 4 q^{9} - 80 q^{11} + 80 q^{15} + 80 q^{19} - 272 q^{21} + 276 q^{25} + 280 q^{29} - 384 q^{31} + 304 q^{35} + 224 q^{39} - 1048 q^{41} + 772 q^{45} + 492 q^{49} - 1920 q^{51} + 1616 q^{55} + 1392 q^{59} - 1384 q^{61} + 608 q^{65} + 112 q^{69} - 1376 q^{71} - 160 q^{75} - 1472 q^{79} - 1484 q^{81} + 1152 q^{85} + 1320 q^{89} - 992 q^{91} + 1456 q^{95} + 3152 q^{99}+O(q^{100})$$ 4 * q - 4 * q^5 - 4 * q^9 - 80 * q^11 + 80 * q^15 + 80 * q^19 - 272 * q^21 + 276 * q^25 + 280 * q^29 - 384 * q^31 + 304 * q^35 + 224 * q^39 - 1048 * q^41 + 772 * q^45 + 492 * q^49 - 1920 * q^51 + 1616 * q^55 + 1392 * q^59 - 1384 * q^61 + 608 * q^65 + 112 * q^69 - 1376 * q^71 - 160 * q^75 - 1472 * q^79 - 1484 * q^81 + 1152 * q^85 + 1320 * q^89 - 992 * q^91 + 1456 * q^95 + 3152 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 2\nu^{3} + 2\nu^{2} + 6\nu ) / 3$$ (2*v^3 + 2*v^2 + 6*v) / 3 $$\beta_{2}$$ $$=$$ $$( -2\nu^{3} + 6\nu^{2} - 6\nu ) / 3$$ (-2*v^3 + 6*v^2 - 6*v) / 3 $$\beta_{3}$$ $$=$$ $$( -4\nu^{3} - 4\nu^{2} + 12\nu ) / 3$$ (-4*v^3 - 4*v^2 + 12*v) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + 2\beta_1 ) / 8$$ (b3 + 2*b1) / 8 $$\nu^{2}$$ $$=$$ $$( 3\beta_{2} + 3\beta_1 ) / 8$$ (3*b2 + 3*b1) / 8 $$\nu^{3}$$ $$=$$ $$( -3\beta_{3} - 3\beta_{2} + 3\beta_1 ) / 8$$ (-3*b3 - 3*b2 + 3*b1) / 8

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$$.

 $$n$$ $$17$$ $$21$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.22474 − 1.22474i −1.22474 − 1.22474i −1.22474 + 1.22474i 1.22474 + 1.22474i
0 6.89898i 0 −10.7980 + 2.89898i 0 12.6969i 0 −20.5959 0
49.2 0 2.89898i 0 8.79796 + 6.89898i 0 16.6969i 0 18.5959 0
49.3 0 2.89898i 0 8.79796 6.89898i 0 16.6969i 0 18.5959 0
49.4 0 6.89898i 0 −10.7980 2.89898i 0 12.6969i 0 −20.5959 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.4.c.c 4
3.b odd 2 1 720.4.f.m 4
4.b odd 2 1 40.4.c.a 4
5.b even 2 1 inner 80.4.c.c 4
5.c odd 4 1 400.4.a.v 2
5.c odd 4 1 400.4.a.x 2
8.b even 2 1 320.4.c.h 4
8.d odd 2 1 320.4.c.g 4
12.b even 2 1 360.4.f.e 4
15.d odd 2 1 720.4.f.m 4
20.d odd 2 1 40.4.c.a 4
20.e even 4 1 200.4.a.k 2
20.e even 4 1 200.4.a.l 2
40.e odd 2 1 320.4.c.g 4
40.f even 2 1 320.4.c.h 4
40.i odd 4 1 1600.4.a.cf 2
40.i odd 4 1 1600.4.a.cm 2
40.k even 4 1 1600.4.a.ce 2
40.k even 4 1 1600.4.a.cl 2
60.h even 2 1 360.4.f.e 4
60.l odd 4 1 1800.4.a.bk 2
60.l odd 4 1 1800.4.a.bp 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.c.a 4 4.b odd 2 1
40.4.c.a 4 20.d odd 2 1
80.4.c.c 4 1.a even 1 1 trivial
80.4.c.c 4 5.b even 2 1 inner
200.4.a.k 2 20.e even 4 1
200.4.a.l 2 20.e even 4 1
320.4.c.g 4 8.d odd 2 1
320.4.c.g 4 40.e odd 2 1
320.4.c.h 4 8.b even 2 1
320.4.c.h 4 40.f even 2 1
360.4.f.e 4 12.b even 2 1
360.4.f.e 4 60.h even 2 1
400.4.a.v 2 5.c odd 4 1
400.4.a.x 2 5.c odd 4 1
720.4.f.m 4 3.b odd 2 1
720.4.f.m 4 15.d odd 2 1
1600.4.a.ce 2 40.k even 4 1
1600.4.a.cf 2 40.i odd 4 1
1600.4.a.cl 2 40.k even 4 1
1600.4.a.cm 2 40.i odd 4 1
1800.4.a.bk 2 60.l odd 4 1
1800.4.a.bp 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 56T_{3}^{2} + 400$$ acting on $$S_{4}^{\mathrm{new}}(80, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 56T^{2} + 400$$
$5$ $$T^{4} + 4 T^{3} - 130 T^{2} + \cdots + 15625$$
$7$ $$T^{4} + 440 T^{2} + 44944$$
$11$ $$(T^{2} + 40 T - 1136)^{2}$$
$13$ $$T^{4} + 5600 T^{2} + \cdots + 6801664$$
$17$ $$T^{4} + 16896 T^{2} + \cdots + 14745600$$
$19$ $$(T^{2} - 40 T - 1136)^{2}$$
$23$ $$T^{4} + 48440 T^{2} + \cdots + 259467664$$
$29$ $$(T^{2} - 140 T - 1244)^{2}$$
$31$ $$(T^{2} + 192 T - 46080)^{2}$$
$37$ $$T^{4} + 75488 T^{2} + \cdots + 314849536$$
$41$ $$(T^{2} + 524 T + 65188)^{2}$$
$43$ $$T^{4} + 90360 T^{2} + \cdots + 576576144$$
$47$ $$T^{4} + 206328 T^{2} + \cdots + 9927332496$$
$53$ $$T^{4} + 624608 T^{2} + \cdots + 72090102016$$
$59$ $$(T^{2} - 696 T - 64752)^{2}$$
$61$ $$(T^{2} + 692 T + 100900)^{2}$$
$67$ $$T^{4} + 52280 T^{2} + \cdots + 565868944$$
$71$ $$(T^{2} + 688 T + 93760)^{2}$$
$73$ $$T^{4} + 621440 T^{2} + \cdots + 9130184704$$
$79$ $$(T^{2} + 736 T + 129280)^{2}$$
$83$ $$T^{4} + 1440440 T^{2} + \cdots + 365991120784$$
$89$ $$(T^{2} - 660 T + 53604)^{2}$$
$97$ $$T^{4} + 478208 T^{2} + \cdots + 26342588416$$